Circular colouring and orientation of graphs (Q1850626)

The author proves that if a graph \(G\) has an orientation \(D\) such that for each cycle \(C\) with \(d|C|\pmod k\in \{1,2,\dots, 2d-1\}\) we have \(|C|/|C^+|\leq k/d\) and \(|C|/|C^-|\leq k/d\), then \(G\) has a \((k,d)\)-colouring and hence the circular chromatic number of \(G\) is at most \(k/d\). The concept of circular chromatic number is a natural generalization of the chromatic number from many different points of view.